On Constant Weight Codes and Harmonious Graphs

Very recently a new method has been developed for finding lower bounds on the maximum number of codewords possible in a code of minimum distance d and length n . This method has led in turn to a number of interesting questions in graph theory and additive number theory. In this brief survey we summarize some of these developments. -/ * Visiting Professor, Computer Science Department, Stanford University, Stanford, California 94305. The preparation of this paper was supported in part by National Science Foundation grant MCS-77-23738 and by Office of Naval Research contract NOOOlk-76-C-0330. Reproduction in whole or in part is permitted for any purpose of the United States government. 1 ON CONSTANT WEIGHT CODES AND HARMOXOUS GRAPHS bY R. L. Graham N. J. A. Sloane Bell Laboratories Murray Hill, N.J. 07974 Introduction Very recently a new method has been developed (see L31, C 5 1 , WI> for finding lower bounds on the maximum number of codewords possible in a code of minimum distance d and length n. This method has led in turn to a number of interesting questions in graph theory and additive number theory. In this brief survey we summarize some of these developments. Background By a code C of length n over a finite field F = GF(q) we mean a subset of Fn, i.e., a set of n-tuples with entries in F. The most common choice for F is GF(2), and we restrict ourselves to this case for the remainder of the paper (although the same techniques apply to all finite fields). In this case C is called a binary code. The minimum distance of C is defined to be